L-space conjecture for detected slopes

Let $M_1$ and $M_2$ be knot manifolds, and let $W=M_1\cup_f M_2$ be their union along their torus boundaries. For $*\in\{LO,CTF\}$, a rational slope is $*$-detected when it has the corresponding order-detection or co-oriented-taut-foliation detection property. An L-space conjecture for detected slopes. If $W$ has property $*\in\{LO,CTF\}$, then the gluing map identifies a rational $*$-detected slope on the boundary of $M_1$ with a rational $*$-detected slope on the boundary of $M_2$. This is stated as a predicted converse to the gluing theorem; the source gives no proof or resolution of the converse.